L(s) = 1 | − i·4-s + (1 − i)7-s + i·9-s + (−1 − i)13-s − 16-s + (1 + i)23-s + (−1 − i)28-s − i·29-s + 36-s − i·49-s + (−1 + i)52-s + (1 + i)53-s + 2i·59-s + (1 + i)63-s + i·64-s + ⋯ |
L(s) = 1 | − i·4-s + (1 − i)7-s + i·9-s + (−1 − i)13-s − 16-s + (1 + i)23-s + (−1 − i)28-s − i·29-s + 36-s − i·49-s + (−1 + i)52-s + (1 + i)53-s + 2i·59-s + (1 + i)63-s + i·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001565417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001565417\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 + iT \) |
good | 2 | \( 1 + iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.45190520524103108025931289373, −9.932294100703347519451800294365, −8.764962612967294719663789221999, −7.61055636225506915729162095210, −7.29137821434148289950849904461, −5.77418450959617869697732827195, −5.05724915634242640488193150188, −4.32330256817179189324365120004, −2.56346500264941068433264239817, −1.25422002409433749954878840117,
2.05841030962527830477507538316, 3.13317389367264562143096942095, 4.38227304064454782608807153750, 5.17543925260232201009803150902, 6.56465872708139332683908208791, 7.23423471355048718072329042542, 8.368381380136127682650206194619, 8.881002273048862580135578209051, 9.623204768969270093328618386830, 11.03587404064170320549590750143